A&A 491, 693-701 (2008)

DOI: 10.1051/0004-6361:20077515

## On surface brightness fluctuations: probabilistic and statistical bases

##### I. Stellar population and theoretical surface brightness fluctuations

**M. Cerviño**

^{1}, V. Luridiana^{1}, and L. Jamet^{1, 2}^{1}Instituto de Astrofísica de Andalucía (CSIC), Camino bajo de Huétor 50, Apdo. 3004, Granada 18080, Spain

e-mail: [mcs;vale]@iaa.es

^{2}Instituto de Astronomía (UNAM), Apartado 70-264, México D.F., Mexico

e-mail: ljamet@astroscu.unam.mx

Received 20 March 2007 / Accepted 1 September 2008

** Abstract ***Aims. *This work aims to provide a theoretical formulation of
surface brightness fluctuations (SBF) in the framework of
probabilistic population synthesis models that have no deterministic
relations between the different stellar components of a population but
only relations on average, and to distinguish between the different
distributions involved in the definition of SBF.*Methods. *By applying
the probabilistic theory of stellar population synthesis models, we
estimate the shape (mean, variance, skewness, and kurtosis) of the
distribution of fluctuations across resolution elements, and examine
the implications for SBF determination, definition and application.*Results. *We distinguish between three definitions
of SBF: (i) stellar population
SBF, which can be computed from synthesis models and provide an
intrinsic metric for fit for stellar population studies; (ii)
theoretical SBF, which include the stellar population SBF plus a
term accounts for the distribution of the number
of stars per resolution element ; theoretical SBF that coincides
with the Tonry & Schneider (1998) definition in the special case
when has a Poisson distribution. We find that the
Poisson contribution to theoretical SBF is around 0.1
stellar population SBF and is negligible;
(iii) observational SBF. We present
alternative ways to compute the SBF and extend the application of
stellar population SBF to defining a metric for fitting for standard
stellar population studies.
*Conclusions. *We demonstrate that SBF are
observational evidence of a probabilistic paradigm in population
synthesis, where integrated luminosities have an intrinsic distributed
nature, and they rule out the commonly assumed deterministic paradigm
of stellar population modeling.

**Key words:**galaxies: star clusters

**--**galaxies: stellar content

**--**methods: data analysis

**©**

*ESO 2008*